Generalised hyperbolic
Parameters	${\displaystyle \lambda }$ (real) ${\displaystyle \alpha }$ (real) ${\displaystyle \beta }$ asymmetry parameter (real) ${\displaystyle \delta }$scale parameter (real) ${\displaystyle \mu }$location (real) ${\displaystyle \gamma =\sqrt{{\alpha }^2-{\beta }^2}}$
Support	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$
PDF	${\displaystyle \frac{(\gamma /\delta {)}^{\lambda }}{\sqrt{2\pi }{K}_{\lambda }(\delta \gamma )}{e}^{\beta (x-\mu )}}$ ${\displaystyle \times \frac{K_{\lambda -1/2}\left(\alpha \sqrt{{\delta }^2+(x-\mu {)}^2}\right)}{{\left(\sqrt{{\delta }^2+(x-\mu {)}^2}/\alpha \right)}^{1/2-\lambda }}}$
Mean	${\displaystyle \mu +\frac{\delta \beta K_{\lambda +1}(\delta \gamma )}{\gamma K_{\lambda }(\delta \gamma )}}$
Variance	${\displaystyle \frac{\delta K_{\lambda +1}(\delta \gamma )}{\gamma K_{\lambda }(\delta \gamma )}+\frac{{\beta }^2{\delta }^2}{{\gamma }^2}\left(\frac{K_{\lambda +2}(\delta \gamma )}{K_{\lambda }(\delta \gamma )}-\frac{K_{\lambda +1}^2(\delta \gamma )}{K_{\lambda }^2(\delta \gamma )}\right)}$
MGF	${\displaystyle \frac{e^{\mu z}{\gamma }^{\lambda }}{{\sqrt{{\alpha }^2-(\beta +z{)}^2}}^{\lambda }}\frac{K_{\lambda }(\delta \sqrt{{\alpha }^2-(\beta +z{)}^2})}{K_{\lambda }(\delta \gamma )}}$

Thegeneralised hyperbolic distribution (GH) is acontinuous probability distribution defined as thenormal variance-mean mixture where the mixing distribution is thegeneralized inverse Gaussian distribution (GIG). Itsprobability density function (see the box) is given in terms ofmodified Bessel function of the second kind, denoted by${\displaystyle K_{\lambda }}$.^[1] It was introduced byOle Barndorff-Nielsen, who studied it in the context of physics ofwind-blown sand.^[2]

This class is closed underaffine transformations.^[1]

Barndorff-Nielsen and Halgreen proved that the GIG distribution isinfinitely divisible and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is thegeneralized inverse Gaussian distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinitely divisible as well.^[3]

An important point about infinitely divisible distributions is their connection toLévy processes, i.e. at any point in time a Lévy process is infinitely divisibly distributed. Many families of well-known infinitely divisible distributions are so-called convolution-closed, i.e. if the distribution of a Lévy process at one point in time belongs to one of these families, then the distribution of the Lévy process at all points in time belong to the same family of distributions. For example, a Poisson process will be Poisson-distributed at all points in time, or a Brownian motion will be normally distributed at all points in time. However, a Lévy process that is generalised hyperbolic at one point in time might fail to be generalized hyperbolic at another point in time. In fact, the generalized Laplace distributions and the normal inverse Gaussian distributions are the only subclasses of the generalized hyperbolic distributions that are closed under convolution.^[4]

As the name suggests it is of a very general form, being the superclass of, among others, theStudent'st-distribution, theLaplace distribution, thehyperbolic distribution, thenormal-inverse Gaussian distribution and thevariance-gamma distribution.

• ${\displaystyle \mathrm{G}\mathrm{H}(-\frac{\nu }{2},0,0,\sqrt{\nu },\mu )}$ is aStudent'st-distribution with${\displaystyle \nu }$ degrees of freedom.
• ${\displaystyle \mathrm{G}\mathrm{H}(1,\alpha ,\beta ,\delta ,\mu )}$ is ahyperbolic distribution.

• ${\displaystyle \mathrm{G}\mathrm{H}(-1/2,\alpha ,\beta ,\delta ,\mu )}$ is anormal-inverse Gaussian distribution (NIG).
• ${\displaystyle \mathrm{G}\mathrm{H}(\text{?},\text{?},\text{?},\text{?},\text{?})}$normal-inverse chi-squared distribution
• ${\displaystyle \mathrm{G}\mathrm{H}(\text{?},\text{?},\text{?},\text{?},\text{?})}$normal-inverse gamma distribution (NI)
• ${\displaystyle \mathrm{G}\mathrm{H}(\lambda ,\alpha ,\beta ,0,\mu )}$ is avariance-gamma distribution
• ${\displaystyle \mathrm{G}\mathrm{H}(1,1,0,0,\mu )}$ is aLaplace distribution with location parameter${\displaystyle \mu }$ and scale parameter 1.

It is mainly applied to areas that require sufficient probability of far-field behaviour^{[clarification needed]}, which it can model due to its semi-heavy tails—a property thenormal distribution does not possess. Thegeneralised hyperbolic distribution is often used in economics, with particular application in the fields ofmodelling financial markets and risk management, due to its semi-heavy tails.

• Continuous distributions

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from January 2018